We now describe the skeleton decomposition of \(Q\) , which encodes the structure of every \(\mathcal {H}^\text{tr}_Q(r)\)  using a forest of plane trees and a collection of truncated quadrangulations. This decomposition was first described in [1]}; see [2]} or [3]} for a more detailed explanation that is compatible with our notations.
Let \(Q^{\prime }\)  be the following modification of \(Q\) : split the root edge of \(Q\)  into a face of degree two, add a loop inside this face that is incident to the root vertex of \(Q\) , and root \(Q^{\prime }\)  at the added edge so that the face of degree 1 lies on the right of the new root edge, see Figure REF . Let \(\partial _0 Q\)  be the cycle of \(Q^{\prime }\)  made of the single root edge of \(Q^{\prime }\) .
For every \(r>0\)  and every edge \(e\)  of \(\partial _r Q\) , \(e\)  splits a face \(f\)  of \(Q^{\prime }\)  into two triangular faces; the one that is contained in \(\mathcal {H}^\text{tr}_Q(r)\)  is called the downward triangle with top edge \(e\). The downward triangles cut the part of \(Q^{\prime }\)  outside \(\partial _{0} Q\)  into a collection of slots, filled with finite maps (possibly reduced to a single edge), see Figure REF . Every slot contained between \(\partial _{r-1} Q\)  and \(\partial _r Q\)  is incident to a unique vertex \(v\)  of \(\partial _r Q\) : we say that the slot is associated with the edge of \(\partial _r Q\)  with tail vertex \(v\) , where the edges of \(\partial _r Q\)  are oriented clockwise.
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